Smooth optimization on manifolds naturally generalizes smooth optimization in Euclidean spaces in a manner that is of interest in a variety of applications, including modal analysis, blind source separation (via independent component analysis), pose estimation in computer vision, and model reduction in dynamical systems. Manifolds of interest include the Stiefel manifolds and Grassmann manifolds.

After presenting a number of motivating applications, we will introduce the basics of differential manifolds and Riemannian geometry, and describe how methods in optimization, like line-search, Newton's method, and trust region methods can be generalized to the case of manifolds. The course will assume only a basic knowledge of matrix algebra and real analysis.